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<h4 class="heading"><span class="type">Paragraph</span></h4>
<ol class="decimal">
<li>
<p>Transform (<code class="code-inline tex2jax_ignore">[cross-reference to target(s) "Hsys" missing or not unique]</code>) to get</p>
<div class="displaymath process-math" data-contains-math-knowls="">
\begin{equation*}
(s^2+3s+2)X(s) = \frac{1}{s},
\end{equation*}
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<div class="displaymath process-math" data-contains-math-knowls="">
\begin{equation*}
X(s) = \frac{1}{s}\cdot\frac{1}{(s+2)(s+1)}=
\frac{1}{s}\left(\frac{-1}{s+2}+\frac{1}{s+1}\right).
\end{equation*}
</div>
</li>
<li>
<p>Apply the Integration property to obtain</p>
<div class="displaymath process-math" data-contains-math-knowls="">
\begin{equation*}
\begin{aligned}
x(t)&amp;=&amp;\int_0^t \left(  -e^{-2\theta}+e^{-\theta} \right) \, d\theta\\
&amp;=&amp; \left[\frac{1}{2}e^{-2\theta}-e^{-\theta} \right]_0^t\\
&amp;=&amp;\frac{1}{2}e^{-2t}-e^{-t} -\frac{1}{2}+1\\
&amp;=&amp;\frac{1}{2}e^{-2t}-e^{-t} +\frac{1}{2}.
\end{aligned}
\end{equation*}
</div>
</li>
<li><p>Show that the solution <span class="process-math">\(x(t)\)</span> is asymptotic to <span class="process-math">\(x=\frac{1}{2},\)</span> why is this obvious as a steady state solution?</p></li>
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<span class="incontext"><a href="sec8_5.html#p-489" class="internal">in-context</a></span>
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